Variant 9.3.4.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. We will say that $f$ is an epimorphism if it is a monomorphism when viewed as a morphism of the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$: that is, if the induced map
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, C ) \xrightarrow { \circ [f] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,C ) \]
induces a homotopy equivalence of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,C)$ with a summand of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,C)$, for each object $C \in \operatorname{\mathcal{C}}$. We will generally avoid this terminology, to avoid confusion with the notion of quotient morphism which we introduce in ยง10.3.2 (see Warning 10.3.2.10).